Levi Civita body

The Levi-Civita body is a body that was invented by Tullio Levi-Civita . The real numbers or the complex numbers are a sub-body of the Levi-Civita body. The Levi-Civita body finds application in the efficient symbolic calculation of values ​​of higher derivatives of functions.

definition

Basic amount of the body

The basic set of the Levi-Civita body are all functions (or ) that have a left-end carrier . ${\displaystyle x\colon \mathbb {Q} \to \mathbb {R} }$ ${\displaystyle x\colon \mathbb {Q} \to \mathbb {C} }$

notation

• Just as the real numbers are abbreviated with, the Levi-Civita field can be abbreviated with or with , depending on whether the basic set consists of real or complex functions. ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {C}}}$
• If there is in the Levi-Civita body and has a non-empty carrier, then one denotes the minimum of the carrier that exists due to left finiteness. ${\displaystyle x}$ ${\displaystyle \lambda (x)}$
• One writes for or and that . ${\displaystyle x,y\in {\mathcal {R}}}$ ${\displaystyle x,y\in {\mathcal {C}}}$ ${\displaystyle r\in \mathbb {Q} }$ ${\displaystyle x=_{r}y:\Leftrightarrow \forall q\in \mathbb {Q} ,q\leq r,x(q)=y(q)}$

addition

The addition of two elements of the basic set and is defined as follows: ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle (x+y)(q):=x(q)+y(q)}$

The additive inverse is as follows:

${\displaystyle (-x)(q):=-x(q)}$

The zero element is:

${\displaystyle 0(q):=0\in \mathbb {R} }$ or. ${\displaystyle 0(q):=0\in \mathbb {C} }$

multiplication

The multiplication of two elements of the basic set and is defined as follows: ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle (x\cdot y)(q):=\sum _{q_{x},q_{y}\in \mathbb {Q} \atop q=q_{x}+q_{y}}x(q_{x})\cdot y(q_{y})}$

One element

The one element of the Levi Civita body is function

${\displaystyle 1(q)=\left\{{\begin{array}{ll}1&{\mbox{falls }}q=0\\0&{\mbox{sonst }}\end{array}}\right.}$.

Multiplicative inverse

If is an element of the Levi-Civita body, then a multiplicative inverse can be constructed as follows: One chooses , where the smallest number is with and . If the carrier of only contains the 0, then is . Otherwise there is one in the Levi Civita body and one first looks for one with . One defines the sequence by and . Then fulfills the desired property. Then is . Now we find the multiplicative inverse of by . ${\displaystyle z}$ ${\displaystyle {\tilde {z}}(q)=z(q-a)\cdot {\frac {1}{b}}}$ ${\displaystyle a\in \mathbb {Q} }$ ${\displaystyle z(a)\neq 0}$ ${\displaystyle b=z(a)}$ ${\displaystyle {\tilde {z}}}$ ${\displaystyle ({\tilde {z}})^{-1}={\tilde {z}}}$ ${\displaystyle z=y+1}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle (x+1)\cdot (y+1)=1}$ ${\displaystyle (y_{i})_{i\in \mathbb {N} }}$ ${\displaystyle y_{0}=y}$ ${\displaystyle y_{i+1}=-y\cdot y_{i}-y}$ ${\displaystyle x=\lim \limits _{n\to \infty }y_{n}}$ ${\displaystyle ({\tilde {z}})^{-1}=x+1}$ ${\displaystyle z}$ ${\displaystyle z^{-1}(q)={\tilde {z}}^{-1}(q+a)\cdot {\frac {1}{b}}}$

Fixed point theorem

The above definition of the multiplicative inverse results from the proof of the fixed point theorem (see in the first source ), which guarantees that the limit of the sequence exists and fulfills the desired property. The fixed point theorem is as follows: ${\displaystyle (y_{i})_{i\in \mathbb {N} }}$

Be . Let or be the set of elements such that . Be further or a function with the properties ${\displaystyle q_{M}\in \mathbb {Q} }$ ${\displaystyle M\subset {\mathcal {R}}}$ ${\displaystyle M\subset {\mathcal {C}}}$ ${\displaystyle x}$ ${\displaystyle \lambda (x)\geq q_{M}}$ ${\displaystyle f\colon M\to {\mathcal {R}}}$ ${\displaystyle f\colon M\to {\mathcal {C}}}$

• ${\displaystyle f(M)\subset M}$
• ${\displaystyle \exists k\in \mathbb {Q} ,k>0:\forall x_{1},x_{2}\in {\mathcal {R}}}$(or ) ${\displaystyle x_{1},x_{2}\in {\mathcal {C}}}$ ${\displaystyle :\forall q\in \mathbb {Q} :x_{1}=_{q}x_{2}\Rightarrow f(x_{1})=_{q+k}f(x_{2})}$

Then there is exactly one or , so that: ${\displaystyle x\in {\mathcal {R}}}$ ${\displaystyle x\in {\mathcal {C}}}$

${\displaystyle x=f(x)}$

Embedding the real or complex numbers

The following function is used to embed the real or complex numbers in the Levi-Civita body:

${\displaystyle \Pi \colon \mathbb {R} \to {\mathcal {R}}}$ or. ${\displaystyle \Pi \colon \mathbb {C} \to {\mathcal {C}}}$

${\displaystyle \Pi (x)(q)=\left\{{\begin{array}{ll}x&{\mbox{falls }}q=0\\0&{\mbox{sonst }}\end{array}}\right.}$.

Here the single element is mapped from or onto the single element from or . Furthermore, there is a homomorphism with respect to addition and multiplication. Hence, the real and complex numbers can be viewed as the sub-bodies of the Levi-Civita body. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \Pi }$

Order of the real Levi-Civita body

Be or .. They say if and . This makes the Levi Civita body of real functions an ordered body . ${\displaystyle x,y\in {\mathcal {R}}}$ ${\displaystyle x>y}$ ${\displaystyle x-y\neq 0}$ ${\displaystyle (x-y)(\lambda (x-y))>0}$

With this order is the number, for example

${\displaystyle d(q)=\left\{{\begin{array}{ll}1&{\mbox{falls }}q=1\\0&{\mbox{sonst }}\end{array}}\right.}$

less than any positive real number.

The Archimedean axiom is not fulfilled for the Levi-Civita body. For example: ${\displaystyle \forall n\in \mathbb {N} :\epsilon \in \mathbb {R} _{+}\Rightarrow n\cdot d<\epsilon }$

root

With regard to the multiplication defined above , each always has exactly different -th roots. For one there are the following numbers of -th roots of : ${\displaystyle y\in {\mathcal {C}}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle x\in {\mathcal {R}}}$ ${\displaystyle n}$ ${\displaystyle x}$

	n odd	n straight
x negative	1	0
x positive	1	2
x zero	1	1

amount

Levi-Civita solid of real functions

Be . The amount of x is defined by: ${\displaystyle x\in {\mathcal {R}}}$

${\displaystyle |x|:=\left\{{\begin{array}{ll}-x&{\mbox{falls }}x<0\\x&{\mbox{sonst }}\end{array}}\right.}$

Levi-Civita body of complex functions

Let , where is the imaginary number . The amount of x is defined by: ${\displaystyle x\in {\mathcal {C}},x=a+ib}$ ${\displaystyle i}$

${\displaystyle |x|:={\sqrt {a^{2}+b^{2}}}}$

Here, the root is meant in relation to the multiplication of the Levi-Civita body defined above.

Semi-norm

Be . Then one can define the following semi-norm on the Levi-Civita body: ${\displaystyle r\in \mathbb {Q} }$

${\displaystyle \|x\|_{r}=\sup _{q\leq r}|x(q)|}$

, where is the absolute value of the real or complex numbers . ${\displaystyle |\cdot |}$

Topologies

Order topology

Be or . Be ${\displaystyle M\subset {\mathcal {R}}}$ ${\displaystyle M\subset {\mathcal {C}}}$

${\displaystyle O(x_{0},\epsilon ):=\{x\in {\mathcal {R}}:|x-x_{0}|<\epsilon \}}$or . ${\displaystyle O(x_{0},\epsilon ):=\{x\in {\mathcal {C}}:|x-x_{0}|<\epsilon \}}$

For the order topology, one defines an open set, if

${\displaystyle \forall x_{0}\in M:\exists \epsilon >0:O(x_{0},\epsilon )\subset M}$.

This topology has the following properties:

• It turns and into disconnected Hausdorff rooms . ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {C}}}$
• With this definition of open sets, and are not locally compact spaces . ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {C}}}$
• In agreeing this topology of the discrete topology match. ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathcal {R}}}$

Semi-standard topology

Be the semi-norm of the Levi Civita body . Be or . Be ${\displaystyle \|\cdot \|_{r}}$ ${\displaystyle M\subset {\mathcal {R}}}$ ${\displaystyle M\subset {\mathcal {C}}}$

${\displaystyle S(x_{0},\epsilon ):=\{x\in {\mathcal {R}}:\|x-x_{0}\|_{1/\epsilon }<\epsilon \}}$.

For the semi-standard topology, M is defined as an open set, if

${\displaystyle \forall x_{0}\in M:\exists \epsilon >0:S(x_{0},\epsilon )\subset M}$.

This topology has the following properties:

• She makes and to Hausdorff spaces with countable bases . ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathcal {C}}}$
• The topology it defines is restricted to the real or complex numbers of the standard topology .

Derivation

One can define a derivation on the Levi Civita body : ${\displaystyle \partial }$

${\displaystyle (\partial x)(q):=(q+1)x(q+1)}$

The following applies to this derivation:

• ${\displaystyle \partial 0=0}$
• ${\displaystyle \forall x\in {\mathcal {R}},x\neq 0:\lambda (\partial x)=\lambda (x)-1}$

Applications

The Levi-Civita body enables the efficient calculation of higher derivatives of functions such as

${\displaystyle x\mapsto {\frac {\sin(x^{3}+2x+1)+{\frac {3+\cos(\sin(\ln |1+x|))}{\exp(\tanh(\sinh(\cosh({\frac {\sin(\cos(\tan(\exp(x))))}{\cos(\sin(\exp(\tan(x+2))))}}))))}}}{2+\sin(\sinh(\cos(\tan ^{-1}(\ln(\exp(x)+x^{2}+3)))))}}}$.

There is a program based on the Levi Civita body that calculates the value of the 19th derivative of this function at the point 0 within less than a second. Mathematica , on the other hand, needs more than 6 minutes to calculate the value of the 6th derivative of this function at the 0 position.

swell

• Martin Berz: Calculus and Numerics on Levi-Civita Fields . In: Martin Berz, Christian Bischof, George Corliss, Andreas Griewank (eds.): Computational differentiation. Techniques, applications, and tools. Proceedings of the 2nd International Workshop held in Santa Fe, NM, February 12-14, 1996 . 1996, ISBN 0-89871-385-4 , chap. 2 ( Online [PDF; accessed June 6, 2013]).
• Khodr Shamseddine, Martin Berz: The Differential Algebraic Structure of the Levi-Civita Field and Applications . ( Online [PDF; 199 kB ; accessed on August 15, 2013]).